The simplest solution would be to start with three passphrases and symmetrically encrypt with each one. A tool like GPG can do the careful work of the key derivation (converting phrases to cryptographic keys, i.e. long pretty-random numbers) and encryption for you. The weakness of this solution is that you need to know the order the keys were used, so you can decrypt in the reverse order.
So here's an alternate solution which doesn't care about the order, but for which you'll have to write your own code (and while the implementation is reasonably straightforward, any time you're considering writing your own cryptographic code you're probably in a dark and dangerous place, and have probably had too many shots of something strong).